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Krylov Complexity and Dynamical Phase Transition in the quenched LMG model (2312.05321v2)

Published 8 Dec 2023 in quant-ph and cond-mat.stat-mech

Abstract: Investigating the time evolution of complexity in quantum systems entails evaluating the spreading of the system's state across a defined basis in its corresponding Hilbert space. Recently, the Krylov basis has been identified as the one that minimizes this spreading. In this study, we develop a numerical exploration of the Krylov complexity in quantum states following a quench in the Lipkin-Meshkov-Glick model. Our results reveal that the long-term averaged Krylov complexity acts as an order parameter for this model. It effectively discriminates between the two dynamic phases induced by the quench, sharing a critical point with the conventional order parameter. Additionally, we examine the inverse participation ratio and the Shannon entropy in both the Krylov basis and the energy basis. A matching dynamic behavior is observed in both bases when the initial state possesses a specific symmetry. This behavior is analytically explained by establishing the equivalence between the Krylov basis and the pre-quench energy eigenbasis.

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Citations (8)
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Summary

  • The paper demonstrates that long-term averaged Krylov complexity acts as an order parameter for dynamical phase transitions in the quenched LMG model.
  • It employs combined numerical simulations and analytical derivations to compare metrics like IPR and Shannon entropy across Krylov and energy bases.
  • The findings offer practical insights for quantum control protocols and deepen our theoretical understanding of non-equilibrium phase behavior.

Overview of Krylov Complexity and Dynamical Phase Transition in the Quenched LMG Model

In the paper titled "Krylov Complexity and Dynamical Phase Transition in the Quenched LMG Model," the authors investigate the intricate dynamics of quantum systems through the lens of Krylov complexity, particularly within the Lipkin-Meshkov-Glick (LMG) model. This exploration is situated in the context of dynamical quantum phase transitions (DQPTs), a non-equilibrium phenomenon of growing interest in quantum statistical mechanics. The paper deploys Krylov complexity to characterize the evolution of quantum states post-quench and examines the dynamical phases induced by such a quench.

Key Concepts and Methodologies

The paper begins by introducing the notion of complexity in quantum systems, emphasizing the role of the Krylov basis as a measure of complexity. The Krylov basis is particularly vital as it minimizes the spreading of quantum states in the Hilbert space, serving as a preferred framework for analyzing quantum dynamics. The core premise is that the long-term averaged Krylov complexity behaves as an order parameter in the quenched LMG model. This paradigm is assessed by observing how the Krylov complexity distinguishes between the two dynamically emergent phases prompted by a quench.

Numerical and Analytical Investigations

The paper employs both numerical simulations and analytical methods to elucidate the role of Krylov complexity. It explores the inverse participation ratio (IPR) and Shannon entropy in both the Krylov and energy bases. A remarkable observation is that these metrics exhibit congruent dynamic behavior across both bases when the initial state possesses a specific symmetry. This symmetry is elucidated through analytical derivations demonstrating the equivalence between the Krylov basis and the pre-quench energy eigenbasis for specific initial conditions.

Numerical results show that under particular conditions, the long-term averaged Krylov complexity acts as a discriminant between different dynamical phases. In essence, it exhibits non-analytic behavior in line with that of traditional order parameters at critical points, thus providing a novel approach to identifying DQPTs, specifically the DPT-I (dynamical phase transition) associated with symmetry breaking.

Implications of the Research

The implications of this research straddle both theoretical and practical domains. Theoretically, the findings underscore the utility of Krylov complexity as a robust tool for probing quantum phase transitions beyond the conventional metrics. It opens avenues for deeper insights into how complexity measures can elucidate the phase landscape of quantum systems under quench dynamics.

Practically, these insights could influence the development of quantum control protocols and the design of quantum systems harnessed for computational tasks, where understanding dynamic stability and phase behavior is crucial. The paper also sets a foundation for future explorations into different quantum models where Krylov complexity might unveil hidden aspects of quantum dynamics.

Future Directions

The paper hints at several future directions, including exploring the general applicability of Krylov complexity in other quantum models with varying symmetries and interaction types. Further, the relation of Krylov complexity with thermalization processes and its potential role in defining new thermodynamic variables in non-equilibrium quantum systems heralds intriguing possibilities for quantum thermodynamics.

In summary, the paper presents a nuanced exploration into the dynamics of the LMG model using Krylov complexity, demonstrating its efficacy in characterizing DQPTs. This approach not only enriches the understanding of quantum dynamics but also proposes a new lens through which future theoretical and practical frameworks might be developed.

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